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Unformatted text preview: Math 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time. Solution Let V = volume, t = time, and S = surface area. Then dV dt =- kS , for k > 0. Since the volume of the raindrop is given by V = 4 3 πr 3 , where r is its radius, and its surface area is given by S = 4 πr 2 , we can solve for r and substitute to get S = 4 π ( 3 4 π ) 2 3 V 2 3 . Thus, dV dt =- cV 2 3 , for some constant c > 0. 23. The temperature of an object changes at a rate proportional to the difference between the temperature of the object itself and the temperature of its surroundings. The ambient temperature is 70 o F, the rate constant is 0.05 ( min ) − 1 . Write a differential equation for the temperature of the object at any time. Solution Let T be the temperature of the object, and t be time. Then we have dT dt = c ( T- 70) =- . 05( T- 70) degrees F/min. Notice that the coefficient c is negative because the object is cooling. Section 1.2 3 Consider the differential equation dy dt =- ay + b , where both a and b are positive numbers. (a) Solve the differential equation Solution Since dy dt =- ay + b ⇒ dy − ay + b = dt . Now we integrate both sides and solve for y to get integraldisplay...
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This note was uploaded on 05/13/2008 for the course MATH 22B taught by Professor Hunter during the Spring '08 term at UC Davis.
- Spring '08